Final answer:
To find the polynomial p in P3(IR) that meets the given conditions and minimizes the integral, we must find a polynomial with (x + 1) squared as a factor to satisfy p(-1) = 0 and p'(-1) = 0, and then determine the coefficients that minimize the integral of (1 - 5x - p(x)) squared from 0 to 1.
Step-by-step explanation:
The student is asking to find a polynomial p in P3(IR), which stands for the space of polynomials of degree less than or equal to 3 with real coefficients. The requirements are that p(-1) = 0 and p'(-1) = 0. Additionally, the integral from 0 to 1 of the square of the function (1 - 5x - p(x)) needs to be minimized.
To find such a polynomial, one must first consider the conditions given. The condition p(-1) = 0 implies that (x + 1) is a factor of p(x). Because p'(-1) = 0, the derivative of p(x) should also have (x + 1) as a factor, meaning that p(x) has (x + 1)2 as a factor. The general form of p(x) meeting these requirements will thus be p(x) = a(x + 1)2(x + b), where a and b are real numbers to be determined.
The next step is to substitute this form into the given integral and find values for a and b that make the result as small as possible. This typically involves finding the coefficients that correspond to the minimum value of a resulting function after carrying out the integration process.